Non-Enlargeable Operators and Self-Cancelling Operators

The ε-enlargement of a maximal monotone operator is a construct similar to the Brøndsted and Rockafellar ε-subdifferential enlargement of the subdifferential. Like the ε-subdifferential, the ε-enlargement of a maximal monotone operator has practical and theoretical applications. Recently, R. S. Burachik and A. N. Iusem ["On non-enlargeable and fully enlargeable monotone operators", [J. Convex Analysis 13 (2006) 603--622] studied conditions under which a maximal monotone operator is non-enlargeable, that is, its ε-enlargement coincides with the operator. Burachik and Iusem studied these non-enlargeable operators in reflexive Banach spaces, assuming the interior of the domain of the operator to be nonempty. In the present work, we remove the assumption on the domain of non-enlargeable operators and also present partial results for the non-reflexive case.